MinT - Best Known (89, 114, s)-Nets in Base 5

Best Known (89, 114, s)-Nets in Base 5

(89, 114, 504)-Net over F₅ — Constructive and digital

Digital (89, 114, 504)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (32, 44, 252)-net over F₅, using
  - trace code for nets [i] based on digital (10, 22, 126)-net over F₂₅, using
    - net from sequence [i] based on digital (10, 125)-sequence over F₂₅, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 10 and N(F) ≥ 126, using
        the Hermitian function field over F₂₅ [i]
2. digital (45, 70, 252)-net over F₅, using
  - trace code for nets [i] based on digital (10, 35, 126)-net over F₂₅, using
    - net from sequence [i] based on digital (10, 125)-sequence over F₂₅ (see above)

(89, 114, 5133)-Net over F₅ — Digital

Digital (89, 114, 5133)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(89, 114, 5049730)-Net in Base 5 — Upper bound on s

There is no (89, 114, 5049731)-net in base 5, because

1 times m-reduction [i] would yield (89, 113, 5049731)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 9 629652 821358 543106 976878 655571 597965 599598 653081 310265 471505 059453 840307 155025 > 5¹¹³ [i]